1. 8.2 Using the Rule of Inference
Predicate Logic
8.2 Using the Rule of Inference

2. Using Rules of Inference
(x)(Ax → Bx)
(x)(Bx → Cx) / (x)(Ax → Cx)
Ax → Bx , UI
Bx → Cx , UI
Ax → Cx ,3, HS
(x)(Ax → Cx) , UG

3. Using Rules of Inference
(x)(Bx → Cx)
(Ex)(Ax ● Bx) / (Ex)(Ax ● Cx)
Am ● Bm , EI
Bm → Cm , UI
Am , Simp
Bm ● Am , Com
Bm , Simp
Cm ,7, MP
Am ● Cm ,8, Conj
(x)(Ax ● Cx) , UG

4. Using Rules of Inference
(x)(Ax → Bx)
~Bm / (Ex)~Ax
Am → Bm , UI
~Am ,3, MT
(Ex)~Ax , EG

5. Using Rules of Inference
(x)[Ax → (Bx v Cx)]
Ag ● ~Bg / Cg
Ag → (Bg v Cg) , UI
Ag , Simp
Bg v Cg ,4, MP
~Bg ● Ag , Com
~Bg , Simp
Cg ,7, DS

6. Using Rules of Inference
(x)[Jx → (Kx ● Lx)]
(Ey)~Ky / (Ez)~Jz
~Km , EI
Jm → (Km ● Lm) , UI
~Km v ~Lm , Add
~(Km ● Lm) , DM
~Jm ,6, MT
(Ez)~Jz , EG

7. Using Rules of Inference
(x)[Ax → (Bx v Cx)
(Ex)(Ax ● ~Cx) / (Ex)Bx
Am ● ~Cm , EI
Am → (Bm v Cm) , UI
Am , Simp
Bm v Cm ,5, MP
Cm v Bm , Com
~Cm ● Am , Com
~Cm , Simp
Bm ,9, DS
(Ex)Bx , EG

8. Using Rules of Inference
(x)(Ax → Bx)
Am ● An / Bm ● Bn
Am → Bm , UI
Am , Simp
Bm ,4, MP
(Ex)Bx , EG
Bn , EI
Bm ● Bn ,7, Conj

9. Using Rules of Inference
(x)(Ax → Bx)
Am v An / Bm v Bn
Am → Bm , UI
An → Bn , UI
(Am → Bm) ● (An → Bn) 2,3, Conj
Bm v Bn ,5, CD

10. Using Rules of Inference
(x)(Bx v Ax)
(x)(Bx → Ax) / (x)Ax
Bx v Ax , UI
Bx → Ax , UI
Ax v Bx , Com
~~Ax v Bx , DN
~Ax → Bx , DM
~Ax → Ax ,7, HS
~~Ax v Ax , DM
Ax v Ax , DN
Ax , Taut
(x)Ax , UG

11. Using Rules of Inference
(x)[(Ax ● Bx) → Cx]
(Ex)(Bx ● ~Cx) / (Ex)~Ax
Bm ● ~Cm , EI
(Am ● Bm) → Cm , UI
~Cm ● Bm , Com
~Cm , Simp
~(Am ● Bm) ,6, MT
Bm , Simp
~Am v ~Bm , DM
~~Bm , DN
~Bm v ~Am , Com
~Am ,11, DS
(Ex)~Ax , EG

12. Using Rules of Inference
(Ex)Ax → (x)(Bx → Cx)
Am ● Bm / Cm
Am , Simp
(Ex)Ax , EG
(x)(Bx → Cx) ,4, MP
Bm → Cm , UI
Bm ● Am , Com
Bm , Simp
Cm ,8, MP

13. Using Rules of Inference
(Ex)Ax → (x)Bx
(Ex)Cx → (Ex)Dx
An ● Cn / (Ex)(Bx ● Dx)
An , Simp
(Ex)Ax , EG
(x)Bx ,5, MP
Cn ● An , Com
Cn , Simp
(Ex)Cx , EG
(Ex)Dx ,9, MP
Dm , EI (Choosing m because n is taken, and don’t know that D can be n)
Bm , UI (B can be n or m, as both appear above it, but m is convenient to get the conclusion)
Bm ● Dm ,12, Conj
(Ex)(Bx ● Dx) , EG

14. Using Rules of Inference
(Ex)Ax → (x)(Cx → Bx)
(Ex)(Ax v Bx)
(x)(Bx → Ax) / (x)(Cx → Ax)
Am v Bm , EI
Bm → Am , UI
~Am → Bm , DM
~Am → Am ,6, HS
Am v Am , DM
Am , Taut
(Ex)Ax , EG
(x)(Cx → Bx) ,10, MP
Cm → Bm , UI (B taking m is okay, but is C?)
Cm → Am ,12, HS
(x)(Cx → Ax) , UG

15. Using Rules of Inference
(Ex)Ax → (x)(Bx → Cx)
(Ex)Dx → (Ex)~Cx
(Ex)(Ax ● Dx) / (Ex)~Bx
Am ● Dm , EI
Dm ● Am , Com
Dm , Simp
(Ex)Dx , EG
(Ex)~Cx ,7, MP
Am , Simp
(Ex)Ax , EG
(x)(Bx → Cx) ,10, MP
Bm → Cm , UI (Again, ,,,
~Cm , EI
~Bm ,13, MT
(Ex)~Bx , EG